Modelling of fluidized bed reactors—IV

Chmdcol Engineering Science, 1976, Vol. 31, pp. 11634178.

Pergmon

Press.

Printed in Great Britain

MODELLING OF FLUIDIZED BED REACTORS-IV? COMBUSTION OF CARBON PARTICLES ALFRED0 L. GORDON8 and NEAL R. AMUNDSON Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455,U.S.A. (Receioed 12April 1976) Abstract--A model consisting of bubble and emulsion phases is used to describe the combustion of carbon particles in a non-isothermal continuous fluidized bed reactor. The particles are burned by two heterogeneous reactions: oxidation by oxygen and reduction of carbon dioxide. Both reactions produce carbon monoxide which is oxidized by incoming air. The complete set of mass and heat conservation equations are presented. They are simplified for the case in which no oxidation of carbon monoxide is considered within the bubbles and when the rate of the heterogeneous reactions at the surface of the particles is much faster than the rate of diffusion of reactants. Several steady state computations are presented and the effect of some of the operational variables is discussed. In most cases ranges of variables were found for which there are multiple solutions. In particular, it was shown that the interchange coefficients between the bubble and emulsion phase have the strongest effect upon multiplicity. Some sufficient conditions for the existence of a unique steady state are derived. INTRODUCTION

A renewed interest in combustion of coal has resulted in the search for new process developments. Among these, the fluidized bed reactor has been shown to possess some

technical and economical advantages over more conventional combustors. High heat transfer coefficients allow for reduced heat transfer surfaces, hence more compact units. Also it would appear that fluidized bed combustion may have some pollution control potential. In fact, lower temperatures of operation may minimize NO, formation and emission, and SO* can react with limestone added to the bed. Although a steady growth of technical knowledge has taken place largely in the design area, and in pilot plants and large scale experiments, little has been done in the field of mathematical modeling of combustion in fluidized beds. This is understandable since the process involves complex coupled phenomena occurring in the bed, with chemical reactions, heat and mass transfer, particle size reduction, gas and solid flow divisions, etc. Yagi and Kunii[l] devised a model for the continuous combustion of carbon particles in a fluidized bed in which they dealt with the distribution of particle sizes in the bed and its dependence upon the combustion kinetics of a single’ particle. However, only an isothermal bed was treated and no consideration was given to the existence of a dilute phase. Moreover, they studied only the simplest kinetic behavior, namely, the direct formation of carbon dioxide from carbon and oxygen. A later paper by Wen and Wang[2] considered models for single particle-gas reaction systems and the ways to relate them to the process in the bed. However, a macroscopic view of the bed does not emerge clearly from this procedure and it is difficult to evaluate the performance of the system. The bed was

assumed to be isothermal and no specifics about the kinetics of the combustion were included. More recent papers by Avedesian and Davidson[3] and Davidson[4] recognized clearly the interdependence of the phases and, in particular, the strong effect that mass transfer between bubbles and emulsion phase and diffusion of reactants to the surface of the particles has upon the whole process. They also considered a much more realistic mechanism of combustion of carbon particles by including the two heterogeneous reactions, the combustion of carbon and reduction of carbon dioxide, as well as the homogeneous oxidation of carbon monoxide. On the other hand they assumed an isothermal bed under batch operation. In this work an attempt is made to develop a mode1 including some features neglected or simplified in previous studies although some assumptions and simplifications will be included as well. In what follows we consider the combustion of carbon particles in a two phase, non-isothermal and continuous fluidized bed reactor. This paper will be followed by a second in which a more genera1 problem will be attacked. THEMODEL

We consider a model, developed chiefly by Davidson and Harrison[5] which assumes the reactor is divided into two phases: a dilute or bubble phase and an emulsion or dense phase, the latter consisting of particles and interstitial gas. The basic assumptions of the theory are (A) the dense phase has the same voidage as at incipient fluidization (though this point is not easy to define with precision) and (B) the flow above that for incipient fluidization passes through as bubbles which are free of particles, uniform in size and distributed uniformly throughout the bed. It is also assumed that there is a crossflow or interchange of heat and mass between the two phases. With these postulates the expansion of the tPrevious papers in this series appearedin this journal: 1974 bed can be determined as follows 29 1173; 197530 847; 197530 1159.

*Present address: Department0 de Ingenieria Quimica, Casilla 53-C, Universidad de Conception, Chile. CES VOL. 31. NO. 12-E

qD = q - qmf= Adua - umf)= uDAd.

(1)

A. L. GORDON and N. R. AMUNDSON

1164

From the assumption of constant voidage in the emulsion phase

v, = cn~v*fand

VP= (1- e,,,,)V,,+

(2)

Similarly the expansion of the bed can be calculated directly from the postulates as

v,=sv,=v,,

then the mass consumption in the interval r to r + Ar per unit time is

8 ( 1

where the solid particles have been assumed to be spherical. With this result, the mass balance, stated previously in words, can be put into symbols, namely:

I_s

Fopo(P

or equivalently

+ ( WP~W(f)h

-(wpb(F)R(P))r+br+31Q(iF)R(i)Ar=0. (4)

From (1) it follows that S can be determined if we have an independent value for the absolute bubble gas velocity. Davidson and Harrison[5] suggested the following relation: +l&r

UD =uo-umf

(9

with r&r= 0.711 (&)“2.

solids leaving in carryover solids “growing” into +

solids “growing” out of the interval to a larger size I

solid consumption due to combustion within interval

=

0.

(7)

(11)

The entrainment of solids is a function of particle size and for each size is represented by the elutriation constant K(r). Levenspiel et a[.[61 represent that constant by: K(r)

F&r)

=

(12)

Wb (r)

where K(I) can be correlated with fluidizing and bed parameters (i.e. Wen and Hashinger [7]). Also, because of the complete backmixing assumption pb (I)= pdr).

(6)

From these equations it is clear that for a given value of the equivalent bubble diameter one can compute volumes and volume fractions of all phases, velocities and flows as well as the expansion of the bed. Carbon particles which are fed to the bed at a constant rate with a uniform temperature are not only consumed by the combustion process but they may leave the bed as outflow (say with the ash) or as carryover entrainment. Under steady state conditions a constant weight of solids will remain in the bed. These flows can be easily visualized. In this system particles will shrink and lose mass due to the combustion. We will neglect the ash in this study and assume that the reactions which take place are all at the surface of the particle with no intraparticle reaction. Thus the particle density remains constant. The dense phase is supposed to be well mixed so that the particle size distribution in the bed will be the same as that in the overflow. A steady state mass balance of particles with sizes between r and r + Ar, per unit time, can be stated as follows:

+

- F,p,(P)Ar - F&P)Ar

(13)

With these relations substituted into (ll), one gets dPb (r)

~+L?(rh(r)=WR(r)

Fopdr)

(14)

where g(r) =

F,/W+K(r)tR’(r)

R(r)

3 -Y*

(144

An initial condition for the differential eqn (14) can be represented as: pb(r > &) = 0, for shrinking particles pb (r CR,,,) = 0, for growing particles I ’

Wb)

Equation (14) represents the general differential equation for the particle size distribution in a fluidized bed. Integration gives

(15) In this work, although it is not essential, a feed of uniform size will be considered with radius Ri. In this case the size distribution of the feed can be represented by (note Ri = RY) PO(r) = S(r - Ri),

(8 = delta function).

(16)

Hence substituting (16) into (15) gives

If R(r) =s

dr

and

N(r) =

wpb(P)Ar ps+nF3

(8) (9)

-8 pb(r) = WR(R~)

e-sh@(SMf

(17)

Modellingof fluidized bed reactors IV

1165

and replacing g(r) by its definition results in

react with the oxygen. Exactly where the oxygen and carbon monoxide react is open to question. Whether the F. r’ carbon monoxide reacts adjacent to the carbon surface, in ’ FI/W+K(~)~~ -PbW= - WR(r)R’exP - R, the boundary layer, or well out in the interstitial gas in the 1 R(r) (18) dense phase or dilute phase probably depends on the conditions in the bed. This question is being investigated which is the size distribution function of particles within and will be dealt with at another time. There are two the bed. For the system described in this work we will limiting cases. The CO burns at the surface or it burns further assume that no particles are removed as overflow homogeneously in the gas phases. We will assume the from the system and hence F, = 0 and no carryover or latter, recognizing, however, that it may be incorrect. We entrainment of particles is allowed so that F2 = 0 or will assume that both heterogeneous reactions (22) and equivalently K(r) = 0. It should be pointed out that in (24) are first order with respect to the gas concentration practice some particles may be elutriated from the and the homogeneous reaction (23) as second order. system, particularly at high values of the superficial Hence: velocity. However, it will be assumed that the bed is provided with a cyclone or other separator which returns rl = kJ& (25) such particles to the bed. r2 = ktCo,.&20,~ or rZ= KM Cco.0 (26) With these assumptions, relation (18) reduces to r3 = ~~CCO~,~ (27)

[I

pb(r)‘-

__ FO r3 WR(r)Ri

(19)

and the size distribution may be normalized over the spectrum of radii by

I

4

0

pb(r) dr = 1.

(20)

From (20) and (19), it follows that

which shows that at steady state F,, the particle flow rate, is a constant whose value depends upon the rate of burning of the carbon particles. In general, the value of R(r) will be dependent upon the concentrations and temperatures of the different phases and thus (21) cannot be solved independently but must be solved simultaneously with the conservation equations. However, in simpler models F0 can be solved explicitly as a function of other variables, thus simplifying the computational procedure. We will consider, as stated earlier, that there are three main reactions taking place in the bed. These are

coo, + t02cg)= COZ@,

(23)

cc.,+ CO26,~2coo,.

(24)

The first and third are solid-gas reactions while the second is generally conceded[llO] to be homogeneous. Thus oxygen, carbon monoxide, and carbon dioxide will diffuse through the boundary layers about the particles, the oxygen and carbon dioxide in so that they can react at the surface and the carbon monoxide out so that it can

with the k’s given by expressions of the Arrhenius type[lO, 13,141. Before stating the conservation equations we summarize the main features and assmptions about the model. Additional details can be found elsewhere[U]. Three phases must be considered when writing the mass and energy conservation equations: a bubble or dilute phase formed by uniform sized spherical bubbles devoid of particles, an interstitial gas phase, and the carbon particles suspended in the interstitial gas in the dense phase. The various relationships used among the flows, heat and mass transfer coefficients and crossflow coefficients are summarized in Table 1. One of the purposes of carrying on combustion in fluidized beds is to replace conventional boilers. Thus we will consider a prototype fluid bed shell boiler [16] and the heat delivered by the fluidized bed to the submerged coil is given by

where T,, is the boiling temperature. The dilute or bubble phase is assumed to be in plug flow and the dense phase is assumed to be perfectly mixed. Because of the combustion process particles shrink so at any instant of time the bed contains particles with different residence times thus giving rise to a distribution of sizes. Because of this, although the whole system may be in steady state, each particle is in its own transient state so its radius and temperature vary with time, although all particles are in the same constant surroundings. Particles will be fed to the bed at a constant rate which is a function of the rate of combustion and hence dependent upon the variables of that particular steady state. Furthermore we will assume that particles stay in the system till they are consumed completely with no entrainment or overflow of carbon particles. In general, concentrations of reactants in the gas phases are small and changes in volume and physical properties are assumed negligible. Changes of properties with temperature are also assumed negligible. The two heterogeneous reactiohs described by (22) and

A. L. GORDONand N. R. AMUNDSON

1166

Table 1. Physical parameters

Sunerficial

at

“alocic”

d u

p qf

1.75:

3

P

&)

2

+

150(1-emf) 3

P

d

Kunii and Levenspiel

Emf

=mf

=

d u P (&!?a e)

(17,

3~(P,-P,)P 2

“Grath and sereacfield 1181

ibid

with

with

(Hce)b

>> (H,,c)b

(Hb.)b

-

Nu

hd Cti k

transfer between bed and submerged coil

lieat

far

d y

P

(%c)b

the heat

conduction

through

ibid 8

PU

< 2000

the particles

TEECONSERVATIONEQUATIONS the

above

assumptions

and

considerations

Dilute (bubble) phase

r~+$=

(Kbe)&jr - C;) t azikzC:DC&

(2%

is

assumed to be fast enough so that no temperature gradients occur within the particles.

With

-

P 2

(24) are assumed to be much faster than the diffusion of gas towards the interior of the particles. Thus, those reactions can be assumed to take place at the surface of the particles (i.e. effectiveness factors are zero). Moreover,

. then:

Ci”,=Cy*

at z*=O;

P,C~ZJD$

= (E&e)b (Tf - T$) + (-

TT,=T’*

at z*=O.

the

steady state mass and energy conservation equations for each of the components and phases are given below:

j=1...3

(29a) AHz)kG$C:~

(30) (30a)

Modellingof Auidizedbed reactors IV

Interstitial gas

1167

Table 2. Reference measures and dimensionless variables Reference

measure

Definition

of

dimensionlers

variables

4r(c~*-CB)+(K*,),V,(eX-cg, 4 + I

Cl

- C$)Wp~(r*) dr* + asVlkzC:IC:I

u&(C$

=O;

j=1...3

(31)

qrp,~~(To*-T:)+(Hbr)bVD(~~-T:) t

I

Length:

%

z

- .ig

Time :

“f

e

e* _- “I

P

Z*

Lf

RI

II

a,h,(T,*-T:)Wpb(r*)dr*+h,A,(T*,,-TT) t v,(-AHz)kzC:,Cf

= 0.

Confenrrsfions:

cJ.

_5$; Cl

To*

Tn

=

r

_F

(32) Tempera-

Solid phase Cforeach particle) Concentration at the surface

ture:

Radius:

kj~(Ci:-Cij)t(YljklC:,tcusjk,C:,=O;

co* 1

%

n - D,I,S

T;: i

f?

”

-

D,l,P

i

j=1...3. (33)

Table3. Dimensionless groups

Rate of shrinkage &(PJJ~

= -Ms,(2kG+

kX:,)

(34) H

r*=Ri

at

0*=0.

(“db’f . pgcgul

=-

(34a)

k” R 01 i

Temperature -&p&T,*)

ID

5

= s,k,C:,(-AH,)

=b;

(-AH)

+ s,ksC:,(-AK) t sphp(T: - T:)

Lj

0*=0.

co* =s

M

This couples the conditions in the bed to the allowable inflow of particles required for steady state operation (see eqn 21)

W -=FO

Ri

r

*3dr*

IOR,‘R(r*)

(36)

where

”

=-3 %

w

=5?

=3 *

“I

Ai =k M : co*& SSl , 7

R (r*) = $,

Nj

08

(35a)

Performing equation

c;*

$a*

gg

(35)

M

T,*=TPo* at

=p ;

“9

-Pelf

S2Jj=C3, j=1...3.

rate of shrinking of each particle

(39)

In view of (39) it follows that: [g]

=$l,“[$:;]dz*

(37) (Kbr)bj = (L)b,

and the aij’s are the stoichiometric coefficients when the set of main reactions is written as 2co-o,-2c 2co*-o,2co zco-Co&!

=0 =0

i = 1 . . .3.

W)

With the above assumptions the number of dimensionless groups is reduced since (see Table 3) oj

=D;

KDIj

=KDI;

Aj

=A;

j = 1...3.

(41)

(38)

= 0.

In order to obtain a set of dimensionless equations we introduce the reference measures of Table 2 and the dimensionless groups of Table 3. Further we assume that the molecular diffusion coefficients of the three reactants are the same so

Finally, we simplify the notation of Table 2 by letting: C,, = Xj; CjD= yj ; j = 1 . . .3 TI = x4;

TD = ~4.

(42)

Thus, we obtain the following set of dimensionless equations:

1168

A. L. GORDONand N. R. AMUNDSON

Dilute (bubble) phase -~+~~~U(xi-~i)+a~jK2Uexp(-N21~4)~1~2=0,

. . . 3 (43)

j=l Yi(0) = Yi” -g

+ &U(X4

- y4) t IGLU

(43d

exp (-NZ/y4)yIyZ = 0

The complete solution of the set of eqns (43)-(S) is an involved process requiring a considerable amount of computing time; however a realistic simplification results when the diffusion of reactants from the bulk phase to the particles is limiting, that is, when the rate of mass transfer of O2 and CO2 is much slower than the rate of their reaction at the surface of the carbon particles (i.e. van der Held[l9]; Field et al.[201). From eqn (33) it follows that

(4) y4(0)= 1.

I

k&:

Wd

(52)

CE=k,tk,P

Interstitial gas ’ (G -xi)pb(r)

y/-Xi t&D(~j_Xj)t3W~D

r2

I II

+ azjKz exp (-NZ/X~)XIX~ 1 - x4 + d&D (14 - x4) t 3 wsyp

= 0, j = 1. . .3 ’

(53)

dr

VP(r) -x&b(r)

I0

(45)

If the rate of reaction at the surface is much faster than the rate of mass transfer

dr

rz

k,Bk,,

t &(T,, -x4) + K&Z exp (- NZ/x4)xIx2= 0.

(46)

and

k,+ k,,

(54)

and then

Solids (for each particle) Concentrations of reactants andproducts at the surface‘ xi - Ci, + aliK,r exp (-NJT,)C,, t ayKsr exp (-N&r,)&

j=l

= 0, . . .3. (47)

Rate of shrinkage dr R(r)=s=-KIDMexp(-N,/T,)C,, - K3DM exp (-N~/T,)G, r(0) = 1.

(48)

This approximation has been used commonly in combustion of carbon particles (i.e. Lowry[21]; Avedesian and Davidson [31). On the other hand, CO (j = 2) is produced by both heterogeneous reactions at the surface of the carbon particles, hence its concentration at the surface will not be negligible. Since it was assumed that molecular diffusion coefficients for 02, CO and COt in the mixture were the same it follows from eqn (33) that

Wa)

c*

=

2k,C:, t2ksC:,

2s

Temperature dT, I 3 T &= 3DLKl d0 r PdO

+

*I

(49)

T,(O) = T;.

(49a)

b = T = k,,

for all j,

c,” = c: f 2(CI: + c:, ‘r’dr 0 R(r)

(50)

dr*=__K dtP

’ y(z) dz. I0 -

(51)

r*

(59)

B r

m

or in dimensionless form

Analysis of the conservation equations for the case in which diffusion of reactants is a limiting step (no

homogeneous reaction in dilute phase) (Referred to as M-Model).

(58)

With the approximations stated, the expression (34) can be reduced to

while f =

(57)

it follows that

Performing equation

I-

(56)

where the values of Cc and Cz can be replaced by (52) and (53) (with k, % k,, and ks % k,,). Because

exp (-N,/T,)C,, pr

,~DL~K~~XP(-N~/T,)C~,+~~(X~-T,) r2 pr

‘=-

ce

kZP

dr -=__

d0

where

Modellingof fluidizedbed reactorsIV K = y(2C:

(52) and (53) and the assumptions given in (54) we have

and B = DM(XII + Ck)

+ C$)

(61) (62)

Cl, =

are constants since the interstitial gas has been assumed perfectly mixed. From (60) it follows by direct integration that r*= l-2BB.

(63)

Relation (63) is an expression of the so-called “square law” for the combustion of solids and liquid drops[211. With the expression for dr*/d0* thus simplified, the flow of particles F. can be calculated explicitly from eqn (36) to be: F

0

=5WK Rz

w

which uncouples the value of F. from the rate of shrinkage Jt(r*) = dr*/d6*. The homogeneous reaction between O2 and CO inside the bubbles will be neglected. Of course this is common practice in catalytic beds where no particles are considered inside the bubbles. In our case we may have considerable homogeneous reaction inside the bubble and that case is studied elsewhere[l5]. With the above assumption eqns (43) and (44) reduce to g=KoU(Xj-Y,),

j=1...3

(‘3)

= H&(x,

CII K,r exp (-NJ%)

+1

C,I

C’s=Kg

exp (-NJT,)

+1

(73)

(74)

and since Kjrexp(-Nj/T,)%l,

j=l

and 3,

(75)

it follows that dl’,

3T, dr

he+Tds=

3DL;xl I 3DL;x, I y(x4;Tp)~ r pr pr

t76j

On the other hand eqns (69) and (70) can be substituted into relations (45) and (46). This, together with the above assumption gives (Y:-xj)(lt(YP)-3WsDXj

‘9 I0 t a2jK2exp (- NJx~)x~x~= 0, j=l . . .3 (77)

I’

(l-x.$)(l+aQ)+3WSyP

(Tp - x4)pb(r) dr

rz

cl

t & (TVs -x4) t KZLZexp (-NZ/X~)XIXZ = 0.

(78)

The integral in (77) can be solved analytically. Substituting (60) and (64) into (19) we get

(654

Yi(0) = Yi” 2

1169

pb(r*)=x

(W

- y4)

5r*4

or

(79)

pb(r)“5r4

yi(z)=xj+(yiO-xj)exp(-‘UKroz)

and so the integral referred to becomes equal to 5/3. On the other hand we notice that the integral in (78) can be integrated if T, is known as a function of r. This can be obtained by manipulating eqn (76) and R(r) given in (60). (67) It follows that

y4(z) = x4 + (1 -x4) exp (- U&z).

(68)

Wd

y.%(O) =1 which can be integrated analytically to give

DL,x, + DLsx,

-t+(x4-Tp)

-

!‘,=Xj+$(y;-X,),

ID

&=xl++(l-xI) ID

j=l

as.3

P

P

Substituting (67) and (68) into (51) we get (69)

1

a first order, linear differential equation for T,(r). With the initial condition T,(l) = Tpowe get:

(70) Tp(r)

=

Tpor3@-‘)+ [r3@-‘)-

t L,x1+ L3x3)E

(Mphx4

11

Mph(l-E)

where

(81) P=h[l-exp(-LJKm)]

(71)

Q=t[l-exp(-LJI&)].

(72)

Equation (49) for the temperature of a particle can also be modified under the assumption of mass diffusion as the limiting step. From the dimensionless form of relations

with E = A /(2x, t x3). The integral containing Tp(r) in eqn (78) can be integrated. Leaving out the details of the integration we obtain (Y:-xj)(l

+ CUP)t 5aljGlxl+ SasjGIxj t azjK* exp (-NZ/X~)XIXZ = 0 j=1...3,

(82)

1170

A. L.

GORWN and

(1 -x.,)(1 + 0Q) + 5(2G2t Gs)x, + 5(G,+ G&3 t + ZfC(T’,, - x,) t K,L, exp (-iVz/x4)x,xz = 0 (83)

N.

R. AMUNDSON

particles which the bed can bum completely under the calculated steady state conditions.

with G, = W,D,

Gz = W,pMDT;

G,= W,DL,,

G4= W,DL,.

(84)

Thus, the original system of equations has been reduced to a set of algebraic, nonlinear equations in the four variables for interstitial gas, that is, the concentrations x1, x2, x3 and the temperature x4. By manipulation of these equations, (82) and (83) can be reduced to a single equation in one variable.

UNIQUENESS The fact that we are dealing with multiple reactions suggests the possibility of multiple solutions and such were indeed found in the numerical solutions (to be presented later). It is desirable for design purposes to obtain conditions under which a unique steady state can be assured. In this section sufficient conditions for uniqueness are presented for the model just described. Equation (85) can be rewritten as

-c=o

F(x)=a+bx 1

F(x ) = lOGs(Gz+ GA) t GsLz t (5Gz- lOG4- GsLz)x~ t G6

’ u(x1)

(92)

with G4)t GsLz t 1 t oQ t I&T,, 6 b = 5G3- 10G4- GaL2, c = (1 t aQ + HcW2. a = ““(;’

+&[Tm--$$I=0

(85)

with

(93)

Also v(x,), defined in (86) can be rewritten as: I

(86)

and

(94) with

Gs=l+aP,

Ga=GstSGI d

G, = 2Gs(lOG, - Gs), Gs = 2Gs(Gs - 5G1).

(87)

Once xl = Cl1 is obtained from (85) the other variables of the interstitial gas phase can be obtained from

x4=

T, =- N2 U(Xl)

x3 = C3, = 2(Gs - G~xI) x2 =

c21=

(89)

Gs- Gex, K2x1 exp (- Nz/x~’

(90)

Equation (85) for xl may be solved by any numerical method such as the Newton-Raphson method. In particular xi represents the dimensionless concentration of 02 in the interstitial gas and its range is limited to the subinterval [0,11.Thus the intersections of F(xJ with the abscissa represent the steady state values of x1. The dilute phase profiles are computed from eqns (67) and (68). The age profiles of each particle, that is, the radius and temperature as a function of the time spent by the particle in the bed are obtained from eqns (63) and (81). A bound for the time spent by a particle in the bed is obtained directly from eqn (63). When r = 0, we get the burning time of one particle as

-

Kd% G5G6Z'

e =

G,/Gs, f = Gs/G'O.

(95)

A positive argument of the logarithm in (94) gives an a priori bound on the values of x1 which will satisfy (92). From the definitions it can be easily shown that d >O. Thus, the requirement of a positive argument in (94) can be expressed as x4x1 + e)txl -f) < 0

tw

which is satisfied if max(O,-e)
(97)

since it can be shown that e G6- lOG, --=p f G6-5G,

(98)

and from the definitions, G1> 0 and Gs>O, so that -e
1taP I <“
(91)

uw

From eqn (64) we can calculate the mass flow PO of

Let xlw and x~,~,_, be limits of the allowable interval for x1 defined in (100).

&

=1= 2B

1 2DM(2x,

+x,)’

1171

Modelliig of fluidied bed reactorsIV

F(xd >O

From (92) it follows that F’(x,)=

6

u (XI) dxl

and hence a sufficient set of conditions for uniqueness is given by: c du(x,) u (xl) dx,

x~,~~.

(102)

Vx,

a(x,)=ln[dx;(~:,x’)]=O

x,+=-~+~[(l+de)‘+4df~“*

since a one-signed derivative means there can’t be more than one point where F(x)) = 0 provided the function is continuous. It may be shown by direct computation that du(xJ dx,

ww

vx,

(W

xl+ >

(106)

(107)

From the definitions in Table 3 and values in Table 5 there results SG, > 1.21(1t al’)

W)

or replacing the dimensionless groups we obtain 4.125(1-e)DL,,,l+&[l-exP(-L&D)] (l-S)U umfRi=

a

VW

x,zyC+j0x1) -, --QI

(117)

lim F(x,)+ tm. x,-n+(-)

(118)

F’(x,) > 0.

(119)

Also, for b >O

Hence, relations (113), (118) and (119)’clearly imply that no solutions exist in the subinterval (x,,,~,,.;x,+). A better picture can be obtained from Fig. 1. Thus the range of XI reduces to the subinterval (x,+;x,,~,) where condition (103) holds without restrictions. Hence, there the criteria for suthcient conditions on uniqueness reduce to relation (109). Four possible curves of F(x,) are drawn in Fig. 1 for the interval (x,+;x,.~~). For curve 1 condition (102) holds. In curves 2 and 3 that condition does not hold for all x1 but still we have a unique solution since the F(x,I

II

The second equation for sufficient conditions for uniqueness can be rewritten as sign F(x~,~~ f sign F(xd.

(110)

On the other hand lim F(x,) = a + bx~,~,, XL-XIJ0W

(111)

lim F(xJ = a + bx,,up. X’X,,“P

(112)

By definition a > 0 and since we are considering parameters for which b > 0 (see (106)) it follows that

(116)

and

or equivalently SG,(L, - 2L1- Lz) > GsLz.

XlJow

which holds if f > -e. Thus the range of x1 can be separated into two subintervals separated by the point of discontinuity xi+. We notice

it follows that the left hand side of eqn (102) is always greater than zero. Thus a sufficient condition for (102) to be valid is given by b =5Gs-lO&G&>O

(115)

It can be easily shown that

and since c =(l+crQ tlYc)N2>0

(114)

the positive root being

(103)

--0,

(113)

However, in the allowable range (xu,,~, xl,& there is a point of discontinuity x1+for which

++duo

r-#-bb,

as x1+xI,I~~ or

Fii. 1. Determination of steady states.

GORWN

A. L.

1172

and N. R. AMUNDSON

condition described is only sufficient. In curve 4, relation (102)does not hold and three steady states arise. It should be pointed out that relation (109) provides qualitative and quantitative information about the influence of some of the parameters on the uniqueness of the solutions. The only operational variable on the left hand side of (109) is the radius Ri of the particles being fed. Hence small particles in the feed are more likely to produce a unique steady state. The value of the right hand side of (109) depends on a greater number of parameters and in a more complicated way. We note that the criteria for uniqueness are independent of parameters related to the particle phase such as the initial temperature I’,‘,‘,the heats of reaction, energies of activation, etc. This is a logical consequence of assuming that the diffusion of reactants is a limiting step for the rate of combustion. On the other hand the effect of bubble diameter and superficial gas velocity is difficult to determine a priori because several parameters are dependent upon them. However, it has been found by computation that the right hand side of (109) increases as the bubble diameter decreases and/or the superficial gas velocity increases. From relation (109) it is possible to determine a value of the superficial gas velocity ~0,~ such that for a given set of parameters the solution is unique for u0 < uOM.Some numerical examples are presented in Table 4 and Fig. 2. For u0 > uoM there may be multiple solutions and the interval of multiplicity can be determined. Figure 3 is a plot of F(xJ defined in (85). It is observed that for increasing values of the superficial velocity u. the solution is unique, multiple solutions occur and finally a unique solution obtains. The interval of multiplicity can be determined by solving the system

M-Model -

K-L

---

Davidson’s

theory theory

Ri=Jmm /

IO 5 d, - Effective

15 20 25 bubble diometer

(ems)

Fig. 2. Determination of critical superficial velocity.

T#= 300%

Ri = O.OOlm

2 s r, 0 -I -2 010 x,

0.20 0.30 0.40 Dimensionless

:

0.50 0.60$.?0: interstitial”

0.80

0.90

’ ’ concentration

1.00

Fig. 3. Determination of interval for multiple solutions (M model).

(121) b?*>uo*) for which the solutions are multiple. In a similar The roots of the above system will give the interval Table4. Critical value of superticial velocity cas.(*)

dg (m)

RI (m)

"0,M'Umf

uo,M'"mf

Davidson

K-L

0.05

0.005

5.248

3.263

0.10

0.005

5.186

4.869

0.15

0.005

5.140

6.678

0.20

0.005

5.105

0.25

0.005

5.071

0.05

0.003

11.805

7.065

0.10

0.003

11.779

10.980

0.15

0.003

11.756

15.551

0.20

0.003

11.759

20.400

0.25

0.003

11.818

25.425

0.05

o.o01(+)

8.575 10.528

way, we can determine intervals of multiplicity for other parameters by solving F(x,;p)=O (122) i&p)=0

with p the parameter whose interval of multiplicity is to be determined. On the other hand from Table 4 and Fig. 2 we observe the effect of the interchange coefficients for interphase transfer between the dilute phase and the interstitial gas phase. Some of these curves correspond to the cases obtained using the theory of Kunii and Levenspiel[22] for interchange coefficients. The others were obtained using Davidson’s theory (see equations in Table 1). The curves drawn suggest that the effect of the interchange coefficients on the uniqueness of the solution is significant. The area under the curves represent values for which uniqueness is assured. These results, although applied to quite a different system are in qualitative agreement with the results of Bukur [23] who found for catalytic reactions that the range of multiplicity of solutions is smaller with

Modellingof fluid&d bed reactors IV Kunii-Levenspiel son’s.

(K-L)

coefficients than with David-

NUMERICAL CALCULATlONS OF STEADY STATES

Influence of the inflowing gas to the bed The influence of the superficial gas velocity and its temperature is shown in Fig. 4. The curves on this figure are constructed by fixing the inlet temperature I“’ of the gas and plotting the values of the superficial velocity of the gas vs the corresponding steady state temperature in the interstitial gas. Similar graphs could be obtained by plotting other dependent variables for the dense phase or the dilute phase. It is of interest in the design of the fluidized bed as a steam generator to have an estimation of the heat delivered by the bed. From eqn (28) the potential heat per unit volume of expanded bed is given by = ”

z ; 2,000

M-Model

5

Ri= 0.005m

1,800

2

The model contains a large number of parameters so no extensive computations are possible. Instead the influence of a selected number of parameters will be presented and analyzed. Numerical computations were performed for different sets of values of &, Ri, UO,To and TPo.The bubble diameter was varied in the range & = 5-25 cm (5, 10, 15, 20, 25cm). For combustion of carbon rather large particles are used (in comparison with a catalytic tluidized bed reactor) and so feed particles in the range Ri = l5 mm (1,3,5 mm) were considered. For each initial size of particles a value of u,,,~was estimated and the superficial velocity was, in general, varied in the range u0 = 2.5-12u,,,, which is within the region of applicability of the model studied. The temperature of the intluent air was varied in the range To = 300-800°K (300,550,8OO”K)which covers both air at ambient temperature as well as preheated air. The temperature of the particles fed was varied in the range TPo= 300-1000°K. The values of other parameters which are not specifically assigned are indicated in Table 5.

Q

1173

hcAc(Tr Tws) A& * -

E” 1,600 fX! 1,400 .; 1,200

600’

’ 4

2

’ 6

Uo/Umf

’ 8

’ ’ ’ ’ ’ ’ 10 12 14 16 18 20

-Excess

flow

-3

mf

= 0.6 m

MS = 12 Kg mole-l

c; = c; 5 0

q, = 1 Kg eec-1

ic = 4.19 x.103 J Kg-1 x-l

=cr

gs = 1.46 x lo3 .,Kg-l 'K-l

T*

a = 1.14 x lo3 3 !Q -1 *K-l g s; = 6 x 10-5 In2 s=c-1

= 323 'K - 373 OK

-AlI1P 2.21 x 106 3 KS mole -All2= 5.71 I lo8 J r(gm1e

-1 -1

-LlH3= -3.50 x lo8 J Kg mole

-1

El/RK = 1.5 I lo4 'K E2/Rg = 1.2 x lo4 'K

Cmf = 0.4

K3'Kg = 2.5 x104

OK

A

h

m-2 *K-l

= 2.50 J aa-'

q

- 6.75 x 10-2 J -l.ec-L%-1 * wg - 4.15 I10 -3 Kg m-1 .ec-l

E %l ki2

-1 = 1.55 x 10' m see = 3.09 x10* m3 Kg de-laoc

= 0.35 Kg m Pz3 -1

fluidization

This derived variable could be plotted vs u. and To; however, we have chosen to plot TI because it is the variable most commonly measured in experimental work. Other variables, QUincluded, are presented for representative values of the parameters in Table 6. Figure 4 includes the curves for feed particles of size 5mm. This is the only value of Ri studied for which multiplicity was found within the range of uo/u,,,,studied. From relation (109) it is clear that the occurence of multiplicity is quadratically enhanced by particles of larger initial size. As shown in Table 5 uniqueness is assured for u~/u,,,~< 3.263 in all the cases presented. On the other hand both sides of (109) are independent of the temperature p but the value of u,/u,,,f at which multiplicity occurs, as well as the range of values for which it occurs, are strongly dependent upon that temperature. These can be calculated by solving the system described in (120) and (121). From Fig. 4 we observe that the region of multiple solutions decreases as the temperature of entering gas increases and it finally disappears at least for the range of u~/u,,,~considered here. The abrupt change in the dependent variables has been reported in an experimental work on fluidized bed combustion[25] using a propane-air-sand bed. In that work, for increasing values of the flow a change in the

L

c; = 4.8 x 1o-3 Kg Irolem

incipient

Fig.4. Effect of u0and Toon possiblesteadystates.

Table 5. Numerical values of fixed parameters used in computations A, = 6 3 %3 = 1.55 X 10’ m set -1 s = 1.17 m2

over

1

-3

p, = 0.8 x lo3 Kg m

-3

1174

A. L. GORDON and N. R. AMIJND~ON Fig.6. Parametervaluesfor calculationof steadystates case

u,/uti

To

T” P

Fo

TI

% x 10-b

QJFO* 1o’6

‘K

OK

Kg/h

‘K

“W/2

IJouleln?Kg1

%

%

m

mn

z

0.15

1

5.5

550

500

348

887

0.486

5.028

II

0.15

3

5.5

550

500

827

967

0.435

1.894

III

0.15

5

5.5

550

500

880

949

0.394

1.612

I"

0.25

1

5.5

550

500

265

840

0.478

6.494

"

0.25

3

5.5

550

500

674

916

0.419

2.238

VI

0.25

5

5.5

550

500

775

a89

0.368

1.709

"II

0.15

3

5.5

800

300

827

1,022

0.472

2.055 2.103

"III

0.15

3

5.5

800

800

827

1,038

0.483

IX

0.15

3

5.5

800

1,000

827

1,044

0.487

2.120

X

0.15

3

2.5

300

500

598

928

0.700

4.214

XI

0.15

3

7.5

300

500

1,010

871

0.290

1.034

XII

0.15

3

12.5

300

500

1,496

852

0.181

0.436

XIII

0.15

3

2.5

800

500

598

l,Ob5

0.789

4.750

XI"

0.15

3

7.5

800

500

1,010

1,045

0.382

1.362

XV

0.15

3

12.5

800

500

1,496

1,091

0.264

0.635

character of the combustion was reported. However, Fig. 4 shows that the phenomenon is indeed a transition from an ignited to a quenched steady state of the system. Moreover, the transition occurs for values of uo larger than the one which gives a maximum in the curves represented. This maximum represents the equilibrium reached by two dissimilar effects. Increasing values of the superficial velocity enhance the combustion process, hence giving higher temperatures in the bed. On the other hand higher values of the superficial velocity increase the cooling effect that the entering gas flow has upon the system. This latter effect is predominant at high flows placing the bed in a quenched state in which the temperature of the interstitial gas can be calculated from a heat balance between the gas and solid inflows to the bed. This double effect can be seen more clearly by observing that in the curve for To = 300°K the change is sharper, and it occurs for moderate values of U&Q while in the curve for inflowing gas at 800°K the cooling effect is less pronounced so the smoother transition to a quenched state results.

1,700 1,600 M - Model Tp*=5000K To= 300*K

I.400 z e 1300 f’

WUmf

=a.5

‘ij 1,200 % E 1,100 t” 9 1,000 .f 2 2 900 E

’ 000 I= 700

I 3 5 7 9 II 13 15 17 19 2123 dg - Effective bubble diameter

25 (ems)

Fig.5. Effectof dB and Ri on possiblesteadystates.

INFLUENCE OFBUBBLE SIZE Care should be exercised in considering the bubble diameter & as an independent variable. In fact for a given value of u. there is a unique value of the effective bubble diameter. However, this value is not known a priori, and, in general, experiments are required to obtain a useful correlation. Moreover, the value of & resulting from the operating superficial velocity can be substantially altered by the geometry of the bed (baffles, grids, distributor design, etc.). Figure 5 shows that the region of values of ds for which multiple steady states occur is indeed very narrow. Thus for Ri = Smm multiple solutions occur up to dn = 5.75cm, and for larger values there exists a unique solution along the low temperature branch. For dB < 5 cm a branch occurs at a very high temperature in spite of the fact that the bed is being fluidized with relatively cool air (T”=3OO”K). Figure 6 shows that this is due to very extreme values for mass and heat transfer (KD and H, respectively in the graph) for small bubbles, thus making possible a very rapid transfer of reactant from the bubbles

M=Modsl To= 300’ K TV = 500’ K Uo/Umf = 5.5

2 .z .o b

0”

3-

5 -

50

-40:

1: 6

G E 5 i E

2-

I

5 dg-Effective

9

13

17

bubble

21

25

diameter

Fig. 6. Influence of dB and R, upon heat and mass transfer betweenbubblephaseand densephase.

Modellingof fluidized bed reactorsIV

to the sites of reaction, that is, from the interstitial gas to the surface of the particles. From Fig. 5 it is apparent that the temperature of the interstitial gas in the bed, hence the potential heat delivered to a submerged surface, will be lower if larger particles are fed to the system. Physically this might be expected. The completion of the exothermic reactions in the bed depends upon the rate of mass and heat transfer from the interstitial gas to the particles. This rate is proportional to the reciprocal of the radius of the particles and thus, for values of ds greater than a certain value (dependent upon the other parameters) the heat delivered not only decreases with increasing size of bubbles but with increasing size of particles in the feed as well. For smaller bubbles the net effect of Ri is the opposite. In fact, for very small bubbles the rate of mass transfer from a bubble to the interstitial gas is so large that enough oxygen is immediately transferred from the bubbles facilitating the highly exothermic oxidation of CO inthe dense phase. This should offset the slower mass transfer from and to the larger particles. As mentioned before other parameters or combinations of .them can be plotted in studying the steady state behavior of the bed. For design purposes and evaluation of the efficiency we are particularly interested in the performance of the bed as an energy source. On the right ordinate of Fig. 7 we have plotted a ratio which is a measure of the rate of circulation of the particles in the bed. For steady state this ratio cannot be arbitrarily predesigned. The ratio described is related to the burning time of a particle, which in this model is also the time of residence of the particle in the bed. From relations (63) and (64) it follows that F,, 2.5 -=w et

1175

mance of the bed, represented in the graph by C&/F0 is also much better for small particles. However, in the design of a fluidiied bed boiler these improvements should be balanced against the increased cost of grinding and mechanical handling required for smaller particles. In the upper left of the ordinate axis has been plotted U = ul/uD, the ratio between the velocity of gas in the emulsion phase and the velocity of rising bubbles. In fact U is a measure of the amount of entering gas which travels through the bed as bubbles compared with that through the emulsion phase. The net effect will be dependent upon the temperature of the fluidizing air. INFLUENCE OF INTERCEANGE COEFFICEh”lS BETWEEN DILUTE P3ASEAND MTEWWWL GAS PEME

These coefficients play a significant role in the two phase theory of fluidization. A sample of their intkence on the multiplicity of steady states is shown in Figs. 8 and 9. To construct the curves on Fig. 8 we have used different expressions for the interchange coefficients between bubbles and interstitial gas (see Table 2). Curve N is obtained by setting HID = KID = 0. The effect of the crossflow is quite significant. Thus for Kunii and Levenspiel coefficients, multiple steady state solutions occur in a very limited range of bubble sizes. For M=Model To= 300’

N-No K

K-L

interchange Kunii -Lcvenspiel

0 - Davidson’s

theory

Llo/Urnf=6.5

(125)

and thus the circulation rate is proportional to the reciprocal of the burning time, as expected. From Fig. 7 feeds of smaller particles have a smaller circulation rate. At the same time the thermal perfor-

J de-Effective

bubble

diameter

kms)

Fii. 8. Effect of interchange coefficients and dB on possible steady states.

2,100 .05

To= 300°K Tp’= 500-K Uo/Umf : 6.5

g 1,900 r 1,700 2 2 1,500 $ I-” 6 1,300

r c

I

t=

d,-Effective

bubble diameter(cms)

Fig. 7. Effect of d, on performance of the bed.

K-L = K”“ll-!_*ve”lpiet D = Dowdsen’s theory

700 500

dB=O.Osm Ri-a.OOSm

6?K-L

z 1,100 .E z 900 2

cl

L 2

3 Uo/U,f

I

4 :

5

6 7 6 9 IO II 12 flow over incipient fluidiration

Excess

13

Fig. 9. Effect of interchange coefficients and uolu,,,, on possible steady states.

1176

A. L.

GORDON

and N. R.

Davidson’s values we get multiplicity except for values of dB so large that the bed would be clearly in a slugging regime. When no crossflow is considered, the steady state is unique for all values of dB. The results shown in Fig. 8 parallel closely the catalytic case [23] in spite of sharp physical and kinetic differences. This suggests that the value of the interchange coefficients is the predominant factor in determining multiplicity of states. Due to the strong dependence of these coefficients upon the effective diameter of bubbles the final influence relies effectively on the latter parameter. For the same value of dB Davidson’s values for the interchange coefficients are much larger than those of Kunii and Levenspiel. Thus an increase in the rate of interchange enlarges the range of parameters for which multiple solutions are found. From Fig. 9 it can be observed that the influence of the interchange coefficients when the ratio Ju,,,~ is the independent variable is less sharp than in the previous case. Nevertheless, it should be pointed out that the ratio u,,/u,,,, plays a significant role in the values of the interstitial variables.

&fUNDSON

for small bubbles while larger ones tend to maintain a driving force through the entire length of the bed. The profiles show the presence of 02, CO and CO, at the top of the bed. Their relative amounts are functions of several parameters and, in particular, of the ratio uO/kf. The picture is different if one considers only the boundary layer of interstitial gas surrounding one particle of carbon, for the same set of reactions[3]. There, the presence of CO is accounted for only in a very thin reaction zone around the particles. However, Fig. 10, as well as the results obtained for other sets of parameters, show that not only the kinetics, but even more importantly, the flow conditions, given by the ratio u,,/u,,,~,are the main factors in determining the composition of the outflowing gases. This is particularly significant for large values of the ratio since in that case the fraction of bubbles in the bed reaches values of tifty per cent or more. Thus, the composition of the gases at the top is mainly accounted for by the composition of the bubbles there. CONCLUSIONS

It was shown that the complex system of mass and heat, conservation equations reduces to a simple non-linear DILUTE PHASE PROFILES system of algebraic equations when the heterogeneous As stated before, we have assumed the dilute (bubble) reactions taking place at the surface of the particles are assumed much faster than the diffusion of reactants phase to be in plug flow so the concentrations of reactants through the surrounding boundary layers. Detailed and products, as well as the temperature are distributed analysis of the system gives sufhciency criteria for along the height of the bed. Those profiles, for a given set of parameters, are uniqueness of solutions. Uniqueness does not depend included in Fig. 10. By comparing the values of C, (1) and upon parameters related to the particle phase. FurtherTn (1) (values at the top of the bed) with the respective CiI more, it was found that uniqueness is favored by small and TI of the interstitial gas, assumed perfectly mixed, we particles, larger bubble sizes and low values of the can observe that the concentration profiles only approach, superficial gas velocity. An algorithm was developed more or less slowly, the latter values while the which allows the determination of the sub-interval of temperatures equalizein a very narrow inlet region of the uniqueness for each parameter of the system. Numerical ‘bed. This is in agreement with the rapid and uniform examples were presented for the parameter u0 and its critical value was found for which the solution is always transfer of heat generally attributed to fluidized beds. Again, the speed with which mass and heat are unique. It was shown that this critical value is strongly transfered bettween bubbles and interstitial gas is dependent upon Ri, dB and the values of the interchange dependent upon the size of the bubbles, hence the effect coefficients. just described is sharper and it occurs in shorter distances Multiple steady states were obtained for selected M= Model Ri=O.O03m ds =O.OSm T0=300K 1; = 1.000 K lJ./Urnf = 8.5

*

I

Dimensionless

height

of

the

bed

Z

Fii. 10. Bubblephaseprofiles.

Modellig of fluidized

ranges of parameters in good agreement with the analytical predictions. In particular it was found that the system presents three possible steady states, one of these corresponding to a quenched state. In accordance with previous experience the middle one is probably unstable. The influence of several operating variables was studied numerically and it was found that the trends agree with the predictions as well as with the experimental and theoretical findings of others. In this regard it was found that one of the main factors in determining the state of the bed, as well as the multiplicity of the system, is the value of the interchange coefficients between bubbles and emulsion phase. Finally it was concluded that the dilute phase profiles, and in particular the values at the top of the bed (outflowing gases) are strongly dependent upon the ratio Ju,,,,. When this ratio becomes relatively large the percent of bubbles in the bed is so large that the composition of outflowing gases is mainly a function of the ratio mentioned above, not too surprising a result. Acknowledgements-Theresearchwas supportedby the National ScienceFoundation.The authorsare indebtedto Mr.HugoCaram for his aid and counsel. NOTATION

total area for heat transfer total cross section of the bed specific heat of coolant specific heat of gas concentration in dilute phase average over the height of CiD concentration in the interstitial gas concentration at the surface of the particle specific heat of solid particle concentration in the gas feed dimensionless group defined in Table 3 effective bubble diameter diameter of one carbon particle outer diameter of horizontal immersed tube molecular diffusivity of reactant j in gas mixture activation energy for reaction i flow rates of solids: feed, outflow and entrainment dimensionless groups defined by eqns (84) and (87) gravity constant function defined by eqn (14a) dimensionless group defined in Table 3 dimensionless group defined in Table 3 heat transfer coefficient bubble-cloud per unit bubble volume heat transfer coefficient bubble-emulsion per unit bubble volume heat transfer coefficient cloud-emulsion per unit bubble volume heat transfer coefficient bed-submerged coil heat transfer coefficient particle-interstitial gas

1177

bed reactorsIV

IL Kz,K, dimensionless groups defined in Table 3 (kc h mass transfer coefficient bubble-cloud per unit bubble volume

(Kbrh mass transfer coefficient bubble-emulsion

I

per unit bubble volume mass transfer coe5cient cloud-emulsion per unit bubble volume dimensionless group defined in Table 3 dimensionless group defined in Table 3 reaction rate for ith reaction thermal conductivity of gas Arrhenius frequency factor per unit surface (reactions 1 and 3) kt1z Arrhenius frequency factor per unit volume (reaction 2) particlek, mass transfer coe5cient interstitial gas 4 dimensionless group defined in Table 3 Lf height of fluidized bed L mf height of fluidized bed at incipient fluidization M dimensionless group defined in Table 3 MS molecular weight of particles N(r) number of particles within the size interval r to r+dr Ni dimensionless group defined in Table 3 Nu, Nusselt number for heat transfer size distribution of solids: feed, outflow, entrainment and bed QC heat rejected by the bed to the walls and the coil Qo heat delivered by unit volume of bed 4 total volumetric gas flow a flow of water through the coil 4D volumetric flow rate of gas in the dilute phase 4x volumetric flow rate of interstitial gas Qmf volumetric flow of gas at incipient fluidization rate of shrinkage of a particle initial radius of a particle radius of a particle at any instant of time minimum size of particles in the feed maximum of particles in the feed Sherwood number for mass transfer external surface of one particle dilute phase (bubble) temperature average over the height of TD temperature of interstitial gas gas feed temperature temperature of a particle temperatures of particles in the feed temperature of entering water to the coil dimensionless group defined in Table 3 relative bubble rise velocity Ubr absolute dilute phase gas velocity UD UI absolute velocity of the interstitial gas umf minimum fluidization velocity superficial gas velocity ;; volume of expanded bed VP total volume of particles

A. L.

1178

GORWN

volume of dilute phase volume of interstitial gas volume of one particle function defined by eqn (86) weight of particles within the bed dimensionless group defined in Table 3 axial distance measured from the bottom of the bed Greek symbols a dimensionless group defined in Table 3 aii stoichiometric coefficients

dimensionless group defined in volume fraction of bubbles lmf voidage in the emulsion phase heat of ith reaction -hHi elutriation constant u(r) hi dimensionless group defined in CL viscosity of gas P dimensionless group defined in Pe density of gas density of a solid particle h specific surface of one particle o, dimensionless group defined in age of a particle ; burning time of a particle @b dimensionless group defined in age of a particle il burning time of a particle eb

Table 3

AMUNDSON

*

dimensional values (of identical dimen sionless variables) REFERENCJ?.S

HI Yagi S. and Kunii D., Fifth Symposium (International) on Combustion, p. 231 1955. PI Wen C. Y. and Wang S. C., Paper presented at the Second

International Conference on Fluidized Bed Combustion. Ohio 1970. [31 Avedesian M. M. and Davidson J. F., Trans. Instn. Chem. Engrs. 197351 121. [41 Davidson J. F. and Harrison D., Fluidization. AcademicPress,

London 1971. PI Davidson J. F. and Harrison D., Fiuidized Particles. Cambridge University Press, Cambridge 1963. bl Levenspiel O., Kunii D. and Fitzgerald T., Powder Technology 1%8/1%9 2 87.

5

Table 3 Table 3

Table 3

Table 3

Subscripts and superscripts D dilute phase

e g I

and N. R.

emulsion phase gas interstitial gas i reaction i i component j s surface of particle 0 inlet conditions for gas and particles

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